arxiv: 2605.10261 · v1 · submitted 2026-05-11 · 💻 cs.AI · cs.LG

Recognition: 2 theorem links

· Lean Theorem

E-TCAV: Formalizing Penultimate Proxies for Efficient Concept Based Interpretability

Hasib Aslam , Muhammad Ali Chattha , Muhammad Taha Mukhtar , Muhammad Imran Malik , Andreas Dengel , Sheraz Ahmed

Authors on Pith no claims yet

Pith reviewed 2026-05-12 05:16 UTC · model grok-4.3

classification 💻 cs.AI cs.LG

keywords TCAVconcept activation vectorsneural network interpretabilitypenultimate layerefficient approximationmodel debugginginter-layer agreement

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The pith

The penultimate layer of a neural network serves as a reliable proxy for TCAV scores from earlier layers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines sources of instability in TCAV and finds that scores from layers in the final block align closely with those from the penultimate layer. This alignment, combined with the effect of the latent classifier choice, lets a single evaluation at the end replace repeated computations earlier in the network. If the pattern holds, concept-based explanations become practical for larger models and repeated use during training. The work tests the pattern on vision and language tasks across multiple architectures.

Core claim

E-TCAV shows that layers in the final network block produce TCAV scores that strongly agree with the penultimate layer once the latent classifier is fixed, and that directional sensitivities become degenerate at that layer. These observations allow the penultimate layer to act as a fast proxy, yielding approximations whose cost scales linearly with network depth and sample count.

What carries the argument

Penultimate-layer proxy for TCAV, enabled by final-block inter-layer score agreement and latent-classifier stability.

If this is right

TCAV computation time drops linearly as network size or evaluation sample count grows.
Model debugging with human concepts becomes feasible for networks too large for repeated full-layer evaluation.
Real-time or iterative concept-guided training loops can incorporate TCAV checks without prohibitive cost.
Score variance decreases when attention shifts to consistent latent classifier selection rather than layer choice.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Proxy methods like E-TCAV could reduce overhead in other activation-based interpretability tools that currently scan multiple layers.
Extending the tests to attention-heavy transformers would check whether the final-block agreement survives changes in architecture.
Linear scaling opens the door to embedding E-TCAV directly inside training loops for on-the-fly concept alignment checks.

Load-bearing premise

The observed agreement among final-block layers and the degeneracy at the penultimate layer will continue to hold for models and data outside the four architectures and five datasets tested.

What would settle it

A new model or dataset where TCAV scores at the penultimate layer differ substantially from those at earlier layers in the final block would show the proxy does not work.

Figures

Figures reproduced from arXiv: 2605.10261 by Andreas Dengel, Hasib Aslam, Muhammad Ali Chattha, Muhammad Imran Malik, Muhammad Taha Mukhtar, Sheraz Ahmed.

**Figure 2.** Figure 2: Linearly scaling E-TCAV speedup. The x-axis is the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

read the original abstract

TCAV (Testing with Concept Activation Vectors) is an interpretability method that assesses the alignment between the internal representations of a trained neural network and human-understandable, high-level concepts. Though effective, TCAV suffers from significant computational overhead, inter-layer disagreement of TCAV scores, and statistical instability. This work takes a step toward addressing these challenges by introducing E-TCAV, a framework for efficient approximation of TCAV scores, which is based on extensive investigation into three key aspects of the TCAV methodology: 1) the effect of latent classifiers on the stability of TCAV scores, 2) the inter-layer agreement of TCAV scores, and 3) the use of the penultimate layer as a fast proxy for earlier layers for TCAV computation. To ensure a solid foundation for E-TCAV, we conduct extensive evaluations across four different architectures and five datasets, encompassing problems from both computer vision and natural language domains. Our results show that the layers in the final block of the neural network strongly agree with the penultimate layer in terms of the TCAV scores, and the commonly observed variance of the TCAV scores can be attributed to the choice of the latent classifier. Leveraging this inter-layer agreement and the degeneracy of directional sensitivities at the penultimate layer, E-TCAV guarantees linearly scaling speed-ups with respect to the network's size and the number of evaluation samples, marking a step towards efficient model debugging and real-time concept-guided training.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

E-TCAV shows that penultimate layers can proxy for earlier ones in TCAV on the tested models, delivering observed linear speedups, but the agreement is empirical only.

read the letter

The main point is that final-block layers produce similar TCAV scores, so computing at the penultimate layer alone gives a faster approximation while the variance mostly traces back to the choice of latent classifier rather than layer depth. They back this with checks across four architectures and five datasets that mix vision and language problems. That range lets them map the agreement pattern and show the speedup scales with network size and sample count, which is the practical gain for anyone running TCAV on bigger models. The experiments are systematic enough to establish the pattern in those cases and to separate the classifier effect from layer position. The soft spot is exactly what the stress-test note flags: the agreement is shown only on those specific models and datasets, with no argument from network properties or training dynamics that would predict it elsewhere. The abstract uses “guarantees” for the linear speedup, but that is an observation from the runs, not a general result, so the approximation error on untested architectures stays uncharacterized. If the full paper includes tight controls on variance and classifier ablations, that part holds up; the generalization question does not. This is for practitioners who already use TCAV and need it to run faster on large networks. Readers working on efficient interpretability tools will find the layer-agreement results and the multi-domain numbers useful. It deserves a serious referee because the efficiency angle is concrete on the tested ground and the investigation is broader than single-model TCAV papers, even though more architectures or a simple theoretical sketch would strengthen the claim.

Referee Report

3 major / 2 minor

Summary. The paper introduces E-TCAV as an efficient approximation framework for TCAV scores that substitutes the penultimate layer as a proxy for earlier layers. This is motivated by three empirical investigations: the impact of latent classifiers on TCAV score stability, inter-layer agreement of TCAV scores, and the suitability of the penultimate layer as a fast proxy. Extensive experiments across four architectures and five datasets (spanning computer vision and NLP) are used to support claims that final-block layers strongly agree with the penultimate layer, that observed TCAV variance stems primarily from latent classifier choice, and that the proxy yields linearly scaling speed-ups with network size and sample count.

Significance. If the observed inter-layer agreement and proxy validity hold more generally, E-TCAV would offer a practical reduction in TCAV's computational overhead, enabling broader adoption for model debugging and concept-guided training. The manuscript earns credit for its cross-architecture, cross-domain empirical evaluation (four models, five datasets) and for identifying the latent classifier as a source of variance, which is a useful diagnostic contribution. However, the absence of theoretical grounding or error bounds means the significance remains scoped to the tested regimes rather than establishing a general principle.

major comments (3)

[Abstract] Abstract: The assertion that E-TCAV 'guarantees linearly scaling speed-ups with respect to the network's size and the number of evaluation samples' is not supported by a derivation or error analysis; the speedup is reported as an empirical outcome of the penultimate proxy on the four tested architectures, without bounds on the approximation error or conditions under which the linear scaling would break.
[Abstract] Abstract and experimental results: The central justification for the proxy rests on the claim that 'layers in the final block of the neural network strongly agree with the penultimate layer in terms of the TCAV scores,' yet no quantitative agreement metric, statistical significance test, or failure-case analysis is provided to characterize when this agreement holds or the magnitude of introduced error; this directly underpins the efficiency claim and its claimed generality.
[Experimental evaluation] The manuscript attributes 'commonly observed variance of the TCAV scores' to the choice of latent classifier, but the experimental controls for isolating this factor (e.g., fixed random seeds, identical concept sets, variance decomposition) are not detailed enough to rule out confounding effects from layer-specific representation geometry.

minor comments (2)

[Introduction] The title emphasizes 'Formalizing Penultimate Proxies,' yet the body is primarily empirical; a short formal statement of the proxy assumption (e.g., as an equation relating TCAV scores at layer l and the penultimate layer) would clarify the contribution.
[Background] Notation for TCAV scores and the latent classifier (e.g., the directional derivative or concept activation vector) should be introduced consistently with standard TCAV references to aid readers.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment point by point below, with planned revisions to improve clarity, rigor, and completeness of the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract: The assertion that E-TCAV 'guarantees linearly scaling speed-ups with respect to the network's size and the number of evaluation samples' is not supported by a derivation or error analysis; the speedup is reported as an empirical outcome of the penultimate proxy on the four tested architectures, without bounds on the approximation error or conditions under which the linear scaling would break.

Authors: We agree that 'guarantees' overstates the claim in the absence of formal error bounds. The linear scaling follows directly from the reduced computational graph: TCAV at the penultimate layer requires only a single forward pass through the final block for all samples, yielding O(1) cost per sample independent of earlier layer depth. We will revise the abstract to state that E-TCAV achieves linearly scaling speed-ups as demonstrated empirically and via complexity analysis across the evaluated architectures. We will add an explicit complexity derivation in the methods and a limitations subsection discussing conditions (e.g., concept sensitivity concentrated in early layers) where the proxy approximation error may increase. revision: yes
Referee: [Abstract] Abstract and experimental results: The central justification for the proxy rests on the claim that 'layers in the final block of the neural network strongly agree with the penultimate layer in terms of the TCAV scores,' yet no quantitative agreement metric, statistical significance test, or failure-case analysis is provided to characterize when this agreement holds or the magnitude of introduced error; this directly underpins the efficiency claim and its claimed generality.

Authors: We accept that the current presentation relies primarily on visual comparisons. The full manuscript already reports average TCAV score differences across layers, but we will augment this with quantitative metrics including Pearson and Spearman correlations, mean absolute percentage error, and paired statistical tests (Wilcoxon signed-rank) with p-values. We will also add a dedicated failure-case subsection analyzing instances of lower agreement (e.g., particular concepts in NLP models or early final-block layers in vision transformers) and quantify the resulting proxy error magnitude. revision: yes
Referee: [Experimental evaluation] The manuscript attributes 'commonly observed variance of the TCAV scores' to the choice of the latent classifier, but the experimental controls for isolating this factor (e.g., fixed random seeds, identical concept sets, variance decomposition) are not detailed enough to rule out confounding effects from layer-specific representation geometry.

Authors: We will expand the experimental protocol section to specify all controls: fixed random seeds for classifier training and evaluation, identical concept sets and activation extraction procedures across all layers, and the same evaluation sample batches. We will incorporate a variance decomposition (via repeated-measures ANOVA treating layer and classifier as factors) and add an ablation where the latent classifier is held fixed while varying the layer to isolate geometry effects. These additions will be presented in a new supplementary table and figure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on empirical measurements of layer agreement

full rationale

The paper's central contribution is an empirical investigation across four architectures and five datasets establishing inter-layer TCAV agreement in final blocks and attributing score variance to latent classifier choice. E-TCAV is then defined as the practical use of the penultimate layer as proxy, with linear speed-ups presented as a direct consequence of these measured properties rather than any mathematical derivation, fitted parameter, or self-referential definition. No load-bearing step reduces to its own inputs by construction, and no self-citations or uniqueness theorems are invoked to close the argument.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method depends on the empirical observation that final-block layers agree with the penultimate layer; no free parameters, new entities, or unstated mathematical axioms are introduced in the abstract.

axioms (1)

domain assumption Inter-layer agreement of TCAV scores in the final network block generalizes across models and tasks
Claimed after testing four architectures and five datasets; treated as the basis for using the penultimate layer as proxy.

pith-pipeline@v0.9.0 · 5581 in / 1140 out tokens · 34286 ms · 2026-05-12T05:16:49.649363+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1 (Degeneracy of TCAV for Affine Classifiers). ... TCAVC,k = I(wk · vC > 0)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat_equivNat unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

E-TCAV guarantees linearly scaling speed-ups with respect to the network's size and the number of evaluation samples

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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